Exponential Sine Research Articles

A study of the ultrasonic and audible frequency response of commercial loudspeakers for use as parametric acoustic arrays

Parametric acoustic loudspeakers (PAL), also known as parametric acoustic arrays (PAA), are speakers that use ultrasonic waves to generate directive audible sound. They emit an ultrasonic carrier modulated in amplitude by the desired audible signal, and due to non-linear propagation in air, this signal is reproduced in the audible range. Typical applications include audio reproduction for entertainment, but also active noise control and acoustic measurement of material properties. PAA typically consist of an array of piezoelectric transducers set on a surface. However, there are other speakers in the market that, due to their characteristics, could also achieve the PAL effect. This work examines the performance of commercial speakers when reproducing. First, the ultrasonic frequency response of each speaker is examined in free-field conditions using exponential sine sweeps (ESS). This enables searching for potential carrier frequencies to modulate the ESS in the ultrasonic range. The ESS are next modulated in the desired ultrasonic region and emitted through each speaker, obtaining their audible and ultrasonic impulse responses. The results show that some commercial speakers can also operate as PALs, which is interesting from an industrial point of view as it lowers costs and simplifies the assembly process.

Effects of Varying Levels of Background Noise on Room Acoustic Parameters, Measured with ESS and MLS Methods

Typically, background noise of different types and levels is present during the measurement of the impulse response in spaces. The two methods that are, in practice, most frequently used in the measurement of the impulse response, are the exponential sine sweep (ESS), and the maximum length sequence (MLS). This study’s objective was to estimate the impact of background noise (white noise, tonal noise) on the acoustic parameters (T30, EDT, C80, and D50) for ESS and MLS measurements, by introducing artificial background noise, employing an external sound source. For this purpose, measurements were performed with varying levels of external noise (in steps of 2 dB), and the effect was assessed, using the relative error compared to measurements without artificial background noise. According to the findings for white noise (as background noise), in the case of T30 and EDT, the difference between the two methods, as well as the relative error, for the initial levels of added background noise, was small. However, for higher levels of added background noise, there was a sharp increase in the relative error, which was greater for the ESS method, both for T30 and EDT. Regarding C80 and D50, while initially the differences between the ESS and MLS methods were small, cumulatively, as the background noise increased, the relative error increased for both methods, with the ESS method showing the largest error. In the case of tonal noise (as background noise), the results were consistent with those observed in the case of white noise. The study’s findings contribute to a better understanding of the ESS and MLS methods, and suggest the expected relative error of acoustic parameters when various types and levels of background noise are present. Additionally, the study suggests, based on background noise and level, the optimum method to conduct impulse response measurements.

Carrier frequency influence on the audible and ultrasonic fields generated by an omnidirectional parametric loudspeaker excited with exponential sine sweeps

While the parametric acoustic array (PAA) phenomenon has been traditionally exploited for generating focused sound spots, an omnidirectional parametric loudspeaker (OPL) is a sound source that uses the PAA for producing omnidirectional sound fields. An OPL is made of hundreds of ultrasound transducers set on a sphere, each one emitting an ultrasonic carrier wave modulated by an audio signal. Thanks to non-linear propagation effects in air, the air itself demodulates the emitted ultrasonic signal bringing it back to the audible range. The carrier frequency fc is a critical parameter to the efficiency of the OPL and it is investigated in this work. To that goal, the performance of an OPL prototype consisting of 750 ultrasound transducers set on a sphere is evaluated for a large set of fc values, which are selected to be close to the resonance frequency of the transducers. Measurements are carried out in free-field conditions using (modulated) exponential sine sweeps as the excitation signal, which have recently proved useful to obtain a good frequency response, sound pressure level and directivity of the OPL in the audible range. That methodology is extended in this work to not only analyse the audible field but also the ultrasonic one. The measurements reveal that the OPL behaviour is very sensitive to fc, yielding to large variations in the ultrasonic and audible range. Therefore, when selecting an fc value, a balance must be found between maximizing the sound pressure levels in the low frequency range and preserving the sound field omnidirectionality, which is important for room acoustic applications.

Exponential sine chaotification model for enhancing chaos and its hardware implementation

Chaotic systems have been intensively studied for their roles in many applications, such as cryptography, secure communications, nonlinear controls, etc. However, the limited complexity of existing chaotic systems weakens chaos-based practical applications. Designing chaotic maps with high complexity is attractive. This paper proposes the exponential sine chaotification model (ESCM), a method of using the exponential sine function as a nonlinear transform model, to enhance the complexity of chaotic maps. To verify the performance of the ESCM, we firstly demonstrated it through theoretical analysis. Then, to exhibit the high efficiency and usability of ESCM, we applied ESCM to one-dimensional (1D) and multi-dimensional (MD) chaotic systems. The effects were examined by the Lyapunov exponent and it was found that enhanced chaotic maps have much more complicated dynamic behaviors compared to their originals. To validate the simplicity of ESCM in hardware implementation, we simulated three enhanced chaotic maps using a digital signal processor (DSP). To explore the ESCM in practical application, we applied ESCM to image encryption. The results verified that the ESCM can make previous chaos maps competitive for usage in image encryption.

Quasi-anechoic frequency response measurement of underwater acoustic devices using exponential sine sweeps

As part of the development of the MITRE BlueTech Lab, an existing 106 × 40 × 15 ft. underground tank is being repurposed as an underwater acoustic test facility. Here, we report on the development of a procedure to extract anechoic frequency responses from measurements made in the otherwise reverberant environment using exponential sine sweeps. A demonstration of the procedure applied to a small-aperture baffled receiver array is shown, and results are compared to time-gated stepped sine wave measurements. Simple geometric rules to maximize measurement bandwidth while ensuring far-field conditions for both transmitter and receiver array are presented.

Robust selection of clean swept-sine measurements in non-stationary noise.

The exponential sine sweep is a commonly used excitation signal in acoustic measurements, which, however, is susceptible to non-stationary noise. This paper shows how to detect contaminated sweep signals and select clean ones based on a procedure called the rule of two, which analyzes repeated sweep measurements. A high correlation between a pair of signals indicates that they are devoid of non-stationary noise. The detection threshold for the correlation is determined based on the energy of background noise and time variance. Not being disturbed by non-stationary events, a median-based method is suggested for reliable background noise energy estimation. The proposed method is shown to detect reliably 95% of impulsive noises and 75% of dropouts in the synthesized sweeps. Tested on a large set of measurements and compared with a previous method, the proposed method is shown to be more robust in detecting various non-stationary disturbances, improving the detection rate by 30 percentage points. The rule-of-two procedure increases the robustness of practical acoustic and audio measurements.

Relation of Some Known Functions in terms of Generalized Meijer G -Functions

The aim of this paper is to prove some identities in the form of generalized Meijer G -function. We prove the relation of some known functions such as exponential functions, sine and cosine functions, product of exponential and trigonometric functions, product of exponential and hyperbolic functions, binomial expansion, logarithmic function, and sine integral, with the generalized Meijer G -function. We also prove the product of modified Bessel function of first and second kind in the form of generalized Meijer G -function and solve an integral involving the product of modified Bessel functions.

Characterization of an omnidirectional parametric loudspeaker with exponential sine sweeps

An omnidirectional parametric loudspeaker (OPL) is a sound source that is made of hundreds of ultrasonic piezoelectric transducers set on a sphere. Relying on the parametric acoustic array (PAA) phenomenon, each transducer emits an ultrasonic carrier beam modulated by an audible signal. Thanks to non-linear propagation in air, natural demodulation takes place and focused audible beams are obtained. An OPL thus emits hundreds of audible sound beams in all directions. In this work, the performance of an OPL prototype containing 750 transducers set on a sphere of radius 0.125 cm is examined in free-field conditions. The frequency response, sound pressure level (SPL) and directivity are measured using exponential sine sweeps (ESS). Compared to broad band signals such as white or pink noise, the OPL can produce higher SPLs with ESS because the acoustic power concentrates on a single frequency only. This makes the OPL suitable for room acoustic measurements. A conventional dodechaedral sound source is also measured for comparison. Results show that the OPL produces a better omnidirectional sound field above ∼1 kHz, but it performs worse at lower frequencies. Moreover, it presents a flatter frequency response in all directions, although it generates lower SPL values. The Directivity Index (DI) required by the ISO 3382-2 and ISO 16283-1 standards has also been computed for both sources. The OPL cannot fulfill the requirements of the standard at low frequencies. This paper also demonstrates and discusses DI measurements of high resolution both for the new OPL and the traditional onmidirectional speakers.

An exponential integrator sine pseudospectral method for the generalized improved Boussinesq equation

A Deuflhard-type exponential integrator sine pseudospectral (DEI-SP) method is proposed and analyzed for solving the generalized improved Boussinesq (GIBq) equation. The numerical scheme is based on a second-order exponential integrator for time integration and a sine pseudospectral discretization in space. Rigorous analysis and abundant experiments show that the method converges quadratically and spectrally in time and space, respectively. Finally the DEI-SP method is applied to investigate the complicated and interesting long-time dynamics of the GIBq equation.

A Vibrational Technique for In Vitro Intraoperative Prosthesis Fixation Monitoring.

In this paper, a new vibrational modal analysis technique was developed for intraoperative cementless prosthesis fixation evaluation upon hammering. An artificial bone (Sawbones)-prosthesis system was excited by sweeping of a sine signal over a wide frequency range. The exponential sine sweep technique was implemented to the response signal in order to determine the linear impulse response. Recursive Fourier transform enhancement (RFTE) technique was applied to the linear impulse response signal in order to enhance the frequency spectrum with sharp and distinguishable peak values indicating distinct high natural frequencies of the system (ranging from 15 kHz to 90 kHz). The experiment was repeated with 5 Sawbones-prosthesis samples. Upon successive hammering during the prosthesis insertion, variation of each natural frequency was traced. Compared to classical Fast Fourier Transform, RFTE provided a better tracing and enhancement of frequency components during insertion. Three different types of frequency evolving trends (monotonically increasing, insensitive, and plateau-like) were observed for all samples, as confirmed by a new finite element simulation of the prosthesis dynamic insertion. Two main mechanical phenomena (i.e., geometrical compaction and compressive stress) were shown to govern these trends in opposite ways. Follow-up of the plateau-like trend upon hammering showed that the frequency shift is a good indicator of fixation. Alongside the individual follow-up of frequency shifts, combinatorial frequency analysis provides new objective information on the mechanical stability of Sawbone-prosthesis fixation. The proposed vibrational technique based on RTFE can provide the surgeon with a new assistive diagnostic technique during the surgery by indicating when the bone-prosthesis fixation is acceptable, and beyond of which further hammering should be done cautiously to avoid bone fracture.

Automatic damage type classification and severity quantification using signal based and nonlinear model based damage sensitive features

Structural health monitoring (SHM) is an emerging technology designed to automate the inspection process undertaken to assess the health condition of structures. The SHM process is classically decomposed into four sequential steps: damage detection, localization, classification, and quantification. This paper addresses damage type classification and severity quantification issues as classification problems whereby each class corresponds to a given damage type or a certain damage extent. A Support Vector Machine (SVM) is used to perform multi-class classification task. Classically, Signal Based Features (SBF) are used to train SVMs when approaching SHM from a machine learning perspective. In this work, starting from the assumption that damage causes a structure to exhibit nonlinear response, it is investigated whether the use of Nonlinear Model Based Features (NMBF) increases classification performance. NMBF are computed based on parallel Hammerstein models which are identified with an Exponential Sine Sweep (ESS) signal. A study of the sensitivity of classification performance to the noise contained in output signals is also conducted. Dimension reduction of features vector using Principal Component Analysis (PCA) is carried out in order to find out if it allows robustifying the classification/quantification process suggested in this work. Simulated data on a cantilever beam with various damage types and severities as well as experimental data coming from a composite aeronautic plate with various damage severities generated with a unique and original laser process are considered for demonstration. For both application cases, results show that by introducing NMBF, classification performance is improved. Furthermore, PCA allows for high recognition rates while reducing features vector dimension.

Determining the confined optical length of high index vertical Si nanoforest arrays for photonic applications

The structural and the optical properties of different Si nanostructures have been compared. Detailed optical properties of Si nanowires arrays of different optical lengths, fabricated by facile electroless etching technique, have been reported. The theoretical calculation of exponential sine profile at constant λ = 600 nm shows a better explanation in terms of gradient index with optical length for vertical nanowires. The observations signify the possibility of strong light trapping due to an exponential gradient towards the high index along the nanowires and the existence of dense subwavelength features. The optical admittance (Ƶ) shows a strong impact on optical distance (Z) for Z < H, owing to the electromagnetic wave interaction with the nanowires that perceive a different Ƶ at the oblique angle of incidence (AOI). In addition, the experimental reflectance data and the theoretical model for transverse electric and transverse magnetic modes predict that an optical length of 5 μm can exhibit a very low reflectance value. This indicates that the Si nanowires are polarization insensitive over a wide range of AOI (0°–80°). Moreover, Raman spectra showed a very strong light confinement effect in the first order transverse optical band with increasing etching depths. The morphological dependent resonance theory predicts a strong localized light field confinement in the lower wavelength regime for SiNWs. The effect on the strong resonant absorption modes was further correlated with the simulation results obtained by using COMSOL. The obtained results are likely to enhance the maximum absorption of SiNWs for various photonic applications.

Model Completeness for the Real Field with the Weierstrass ℘ Function

AbstractWe prove model completeness for the expansion of the real field by the Weierstrass ℘ function as a function of the variable z and the parameter (or period) τ. We need to existentially define the partial derivatives of the ℘ function with respect to the variable z and the parameter τ. To obtain this result, it is necessary to include in the structure function symbols for the unrestricted exponential function and restricted sine function, the Weierstrass ζ function and the quasi-modular form E2 (we conjecture that these functions are not existentially definable from the functions ℘ alone or even if we use the exponential and restricted sine functions). We prove some auxiliary model-completeness results with the same functions composed with appropriate change of variables. In the conclusion, we make some remarks about the non-effectiveness of our proof and the difficulties to be overcome to obtain an effective model-completeness result, and how to extend these results to appropriate expansion of the real field by automorphic forms.

Comparison of least squares and exponential sine sweep methods for Parallel Hammerstein Models estimation

Abstract Linearity is a common assumption for many real-life systems, but in many cases the nonlinear behavior of systems cannot be ignored and must be modeled and estimated. Among the various existing classes of nonlinear models, Parallel Hammerstein Models (PHM) are interesting as they are at the same time easy to interpret as well as to estimate. One way to estimate PHM relies on the fact that the estimation problem is linear in the parameters and thus that classical least squares (LS) estimation algorithms can be used. In that area, this article introduces a regularized LS estimation algorithm inspired on some of the recently developed regularized impulse response estimation techniques. Another mean to estimate PHM consists in using parametric or non-parametric exponential sine sweeps (ESS) based methods. These methods (LS and ESS) are founded on radically different mathematical backgrounds but are expected to tackle the same issue. A methodology is proposed here to compare them with respect to (i) their accuracy, (ii) their computational cost, and (iii) their robustness to noise. Tests are performed on simulated systems for several values of methods respective parameters and of signal to noise ratio. Results show that, for a given set of data points, the ESS method is less demanding in computational resources than the LS method but that it is also less accurate. Furthermore, the LS method needs parameters to be set in advance whereas the ESS method is not subject to conditioning issues and can be fully non-parametric. In summary, for a given set of data points, ESS method can provide a first, automatic, and quick overview of a nonlinear system than can guide more computationally demanding and precise methods, such as the regularized LS one proposed here.

Sine and cosine equations on hypergroups

This article deals with trigonometric functional equations on hypergroups. We describe the general continuous solution of sine and cosine addition formulas and a so-called sine-cosine functional equation on a locally compact hypergroup in terms of exponential functions, sine functions, and second-order generalized moment functions.

Experimental techniques for the visualization of the acoustic fields radiating from brass musical instruments

Radiation from brass instrument bells under artificial excitation can be visualized using a number of experimental techniques. In this work, we present two different approaches: Schlieren optical imaging using a high speed camera and acoustic 2D scanning using a linear microphone array. The potentials and limitations of both methods, as well as their methodological requirements, are discussed. Examples of measurements on several instruments are presented. Schlieren optical imaging is applicable for the detection of large amplitude wavefronts formed by nonlinear wave steepening and is this is particularly sensitive to the high frequencies that form within these waves. The 2D scanning linear microphone array on the other hand can, in principle, measure all frequencies and the use of the exponential sine sweep method allows any nonlinear behaviour (in the loudspeaker and microphone or in the air) to be removed to get an accurate measurement of the linear radiation field for any frequency of choice within the measurement bandwidth.Radiation from brass instrument bells under artificial excitation can be visualized using a number of experimental techniques. In this work, we present two different approaches: Schlieren optical imaging using a high speed camera and acoustic 2D scanning using a linear microphone array. The potentials and limitations of both methods, as well as their methodological requirements, are discussed. Examples of measurements on several instruments are presented. Schlieren optical imaging is applicable for the detection of large amplitude wavefronts formed by nonlinear wave steepening and is this is particularly sensitive to the high frequencies that form within these waves. The 2D scanning linear microphone array on the other hand can, in principle, measure all frequencies and the use of the exponential sine sweep method allows any nonlinear behaviour (in the loudspeaker and microphone or in the air) to be removed to get an accurate measurement of the linear radiation field for any frequency of choice within the...

Measurement of Loudspeakers with a Laser Doppler Vibrometer and the Exponential Sine Sweep Excitation Technique

Measurement of Loudspeakers with a Laser Doppler Vibrometer and the Exponential Sine Sweep Excitation Technique

Parallel Hammerstein Models Identification using Sine Sweeps and the Welch Method

Linearity is a common assumption for many real life systems. But in many cases, the nonlinear behavior of systems cannot be ignored and has to be modeled and estimated. Among the various classes of nonlinear models present in the literature, Parallel Hammertein Models (PHM) are interesting as they are at the same time easy to understand as well as to estimate when using exponential sine sweeps (ESS) based methods. However, the classical EES-based estimation procedure for PHM relies on a very specific input signal (ESS), which limits its use in practice. A method is proposed here based on the Welch method that allows for PHM estimation with arbitrary sine sweeps (ASS) which are a much broader class of input signals than ESS. Results show that for various ASS, the proposed method provides results that are in excellent agreement with the ones obtained with the classical ESS method.

Automatic Damage Quantification Using Signal Based And Nonlinear Model Based Damage Sensitive Features

Structural Health Monitoring (SHM) can be defined as the process of acquiring and analyzing data from on-board sensors to evaluate the health of a structure. Classically, an SHM process can be performed in four steps: detection, localization, classification and quantification. This paper addresses damage quantification issue as a classification problem whereby each class corresponds to a certain damage extent. Starting from the assumption that damage causes a structure to exhibit nonlinear response, we investigate whether the use of nonlinear model based features increases classification performance. A support Vector Machine (SVM) is used to perform multi-class classification task. Two types of features are used as inputs to the SVM algorithm: Signal Based Features (SBF) and Nonlinear Model Based Features (NMBF). SBF are rooted in a direct use of response signals and do not consider any underlying model of the test structure. NMBF are computed based on parallel Hammerstein models which are identified with an Exponential Sine Sweep (ESS) signal. A study of the sensitivity of classification performance to the noise contained in output signals is also conducted. Dimension reduction of features vector using Principal Component Analysis (PCA) is carried out in order to find out if it allows robustifying the quantification process suggested in this work. Simulation results on a cantilever beam with a bilinear torsion spring stiffness are considered for demonstration. Results show that by introducing NMBF, classification performance is improved. Furthermore, PCA allows for higher recognition rates while reducing features vector dimension. However, classifiers trained on NMBF or on principal components appear to be more sensitive to output noise than those trained on SBF.

Functional equations on hypergroup joins

In this paper we study functional equations on hypergroup joins. The functional equations in question characterize basic function classes, like exponential functions, additive functions, quadratic functions, and sine functions. We describe all solutions of these functional equations.